[Translated by DeepL. 2nd Oct. 2025]



f(x) = x⁷ + (a)x⁵ + (b)x⁴ + (c)x³ + (d)x² + (e)x + (f) (-15 ≤ a, b, c, d, e ≤ 15, 1 ≤ f ≤ 15, a, b, c, d, e, f are integers)

I calculated the number of irreducible and solvable equations using SageMath. The total computation time was approximately two weeks! The results are output in the form [[a,b,c,d,e,f], order]. The order is the order of the Galois group Gal.



When the order is 42, Gal=F42=C7*C6; when the order is 21, Gal=F21=C7*C3; when the order is 14, Gal=D7; when the order is 7, Gal=C7.

However, I'm not confident about this naming convention. Wikipedia refers to groups of order 42 and 21 as “MetaCyclic groups.” In any case, they are both semidirect products of two cyclic groups.



According to Wiki, the order of the Galois group for an irreducible, solvable quintic equation is one of 42, 21, 14, or 7. However, for some reason, within the range I calculated, the orders were only 14 and 42. I was particularly surprised that the cyclic group C7 was absent. Expanding the range might reveal it, but calculating Galois groups for 7th degree equations is very difficult even with SageMath, so I haven't found it yet. The program is as follows. For the actual computation, I split the calculations into 31 parts by substituting “i=-15,-14,,,15” into each part. Of course, I didn't use 31 PCs; I ran 31 Jupyter notebooks on a single PC. Even so, the computation took about a day. When using Mathematica normally, you probably only use a single core. Also, since I haven't done much video editing lately, the PC I replaced a week ago for the first time in 10 years (CPU: Core Ultra7 265K) had been sitting idle. Finally, I was able to reap the benefits of multi-core processing. Note that when copying and pasting to compile this, I might have missed something. If you spot any mistakes, I'd appreciate it if you could let me know.





_.<x>= PolynomialRing (QQ)

for i in range(-15,16):

  for j in range(-15,16):

     for k in range(-15,16):

        for l in range(-15,16):

           for m in range(-15,16):

             for n in range(1,16):

               f=x^7+i*x^5+j*x^4+k*x^3+l*x^2+ m*x+n  

               if f.is_irreducible():

                  g=f.galois_group()

                  if g.is_solvable():

                    print([[i,j,k,l,m,n},g.order()},)



Now, the calculation results are:



F42: 50, D7: 29, Total: 79.



And the number of f(x) satisfying the conditions a to f above is

31^5*15=429,437,265 (approximately 430 million).





Japan's population is about 124 million, and the US population is 340 million. Therefore, the number of “solvable people” is only about 80 when combining Japan and the US, and only about 23 in Japan alone! Moreover, over half of these people live concentrated in three major metropolitan areas. (Since the Gal=F42 functions in these three regions are expected to be mutually translatable with simple transformations, perhaps half of these 80 people are actually relatives living together in large homes spread across the three regions, while the other half live separately in nuclear families?)







{{-7, 0, 14, 0, -7, 3}, 42},

{{-7, 0, 14, 0, -7, 4}, 42},

{{-7, 0, 14, 0, -7, 5}, 42},

{{-7, 0, 14, 0, -7, 6}, 42},

{{-7, 0, 14, 0, -7, 7}, 42},

{{-7, 0, 14, 0, -7, 8}, 42},

{{-7, 0, 14, 0, -7, 9}, 42},

{{-7, 0, 14, 0, -7, 10}, 42},

{{-7, 0, 14, 0, -7, 11}, 42},

{{-7, 0, 14, 0, -7, 12}, 42},

{{-7, 0, 14, 0, -7, 13}, 42},

{{-7, 0, 14, 0, -7, 14}, 42},

{{-7, 0, 14, 0, -7, 15}, 42},

{{-7, 7, 7, -14, 0, 9}, 14},

{{-7, 7, 7, 7, 7, 3}, 14},

{{-7, 9, -1, -9, 7, 7}, 14},

{{-5, -4, 8, 3, 11, 7}, 14},

{{-4, 6, 15, 3, 3, 1}, 14},

{{-3, 2, 7, 6, -3, 1}, 14},

{{-3, 2, 6, -6, -1, 12}, 42},

{{-2, -3, 1, 5, 4, 1}, 14},

{{-2, -3, 8, 2, -12, 7}, 14},

{{-2, -3, 14, -4, 6, 1}, 14},

{{-2, -1, -2, 0, 6, 5}, 14},

{{-2, 1, 4, -1, -4, 3}, 42},

{{-1, 0, 3, -4, -9, 11}, 14},

{{-1, 3, -3, -2, 14, 7}, 14},

{{-1, 5, 3, -5, 3, 7}, 14},

{{0, -8, 10, 8, 12, 2}, 42},

{{0, 0, -7, 14, -7, 8}, 42},

{{0, 0, 0, 0, 0, 2}, 42},

{{0, 0, 0, 0, 0, 3}, 42},

{{0, 0, 0, 0, 0, 4}, 42},

{{0, 0, 0, 0, 0, 5}, 42},

{{0, 0, 0, 0, 0, 6}, 42},

{{0, 0, 0, 0, 0, 7}, 42},

{{0, 0, 0, 0, 0, 8}, 42},

{{0, 0, 0, 0, 0, 9}, 42},

{{0, 0, 0, 0, 0, 10}, 42},

{{0, 0, 0, 0, 0, 11}, 42},

{{0, 0, 0, 0, 0, 12}, 42},

{{0, 0, 0, 0, 0, 13}, 42},

{{0, 0, 0, 0, 0, 14}, 42},

{{0, 0, 0, 0, 0, 15}, 42},

{{0, 0, 7, -7, 7, 1}, 14},

{{0, 4, 1, -2, 2, 3}, 14},

{{0, 6, -3, 6, 1, 12}, 42},

{{0, 7, -14, 0, 0, 7}, 14},

{{0, 7, 7, 7, 14, 9}, 14},

{{0, 7, 14, 14, 7, 2}, 42},

{{1, -4, -1, 0, 5, 1}, 14},

{{1, 9, 2, -1, 11, 4}, 42},

{{2, 0, 3, 5, -7, 5}, 14},

{{2, 4, -5, 7, -3, 1}, 14},

{{2, 6, 6, -2, 1, 2}, 14},

{{3, 1, 2, 7, -2, 4}, 14},

{{4, -11, 13, -3, -14, 13}, 14},

{{4, -2, 2, 0, -3, 2}, 14},

{{4, 4, 1, 12, -2, 1}, 14},

{{5, 3, 3, 10, 2, 1}, 14},

{{6, 6, 1, 7, -5, 1}, 14},

{{7, 0, 7, 0, 7, 3}, 14},

{{7, 0, 14, -14, -7, 6}, 42},

{{7, 0, 14, 0, 7, 1}, 42},

{{7, 0, 14, 0, 7, 2}, 42},

{{7, 0, 14, 0, 7, 3}, 42},

{{7, 0, 14, 0, 7, 4}, 42},

{{7, 0, 14, 0, 7, 5}, 42},

{{7, 0, 14, 0, 7, 6}, 42},

{{7, 0, 14, 0, 7, 7}, 42},

{{7, 0, 14, 0, 7, 8}, 42},

{{7, 0, 14, 0, 7, 9}, 42},

{{7, 0, 14, 0, 7, 10}, 42},

{{7, 0, 14, 0, 7, 11}, 42},

{{7, 0, 14, 0, 7, 12}, 42},

{{7, 0, 14, 0, 7, 13}, 42},

{{7, 0, 14, 0, 7, 14}, 42},

{{7, 0, 14, 0, 7, 15}, 42},

{{9, -3, 1, 9, 4, 1}, 14},